Intermediate Physics
Circular Motion
Spin
“G”-Forces
Circular Motion (1)
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Circular motion is classified as spin or orbital motion.
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Spin is motion of an object around its center of mass
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Orbital is the motion of the center of mass of a particle about an origin.
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Example is a figure skater spinning around his/her center of mass
Example is a person sitting in a roller coaster cart as it travels around a
“circular” loop or a person swinging a ball on a string (both have spin AND
angular motion)
The rate of spin or orbit is called angular speed (ω) and the angular
velocity (ω) is always perpendicular to the plane of motion (parallel
to the axis of rotation). The direction of ω indicates whether the
object is moving clockwise or counter-clockwise and is found using
the right-hand-rule.
We usually measure ω in radians per second and we can calculate
instantaneous or tangential speed v from ω through the relation
v= ω x r where r = radius. In circular motion, this expression
reduces to v=ωr. This will be explained later when the cross product
is discussed.
Circular Motion (2)
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In uniform circular motion angular velocity remains
constant but tangential velocity at a particular point
changes at every instant (not the magnitude, but
direction is continuously changing).
A changing velocity implies a nonzero acceleration
is present. We call this centripetal acceleration
which is defined as:
v2 ˆ
ac  N
R
The direction is always towards the center of the circle
Circular Motion (3)
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Angular Acceleration (α)
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Picture a washing machine starting its spin cycle; it takes time to
get up to speed. Also picture how a cart on a roller coaster track
entering a loop has a higher speed than it has at the top of the
loop - it must be “decelerating.”
Do not confuse angular acceleration with centripetal acceleration.
In the roller coaster case, there are both centripetal acceleration
and angular acceleration - they are very different quantities.
α is derived in a similar way as translational acceleration – take
the time derivative of angular velocity.
d  (t )
dt
You can calculate tangential acceleration by the relation a=αr,
where r = radius.
 (t ) 
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Spin (1)
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When discussing a rolling ball coaster, we are
considering translational motion and spinning, not
orbital motion.
Convince yourself that every point on a spinning
ball, regardless of radius from the axis of spin, is
moving at the exact same angular speed (rad/s).
Recalling the equation for kinetic energy of rotation,
K r = ½Iω2, we can now define all these terms.
Note that the form of K r is similar to that of
translational kinetic energy, K t = ½mv2, except I is
substituted for m and ω for v.
Spin (2)
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We have defined ω as the angular equivalent
to linear velocity. I is, in fact, the angular
counterpart to mass.
Moment of Inertia, I, is a proportionality
constant relating energy to angular speed. It
incorporates distribution of mass.
I is different for different types of objects. For
example, I = ½mr² for a disk, I = mr² for a
ring, and I = (2/5)mr² for a sphere.
“G”-Forces (1)
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A “G”-Force is a way to quantify the sum of all forces on
a person. We can experience both positive and negative
G-forces.
If the sum of all forces acting on a person is the force of
gravity, the person experiences 1 G. So to find the total
number of G-Forces, divide the total force by the
acceleration of gravity (9.81m/s2) times your mass.
The force of gravity when you sit, stand or lie down is 1
G, though the total sum of the forces on you is zero.
If you carry someone of the same weight on your back
and walk around, you’re feeling a force of 2 G’s.
“G”-Forces (2)
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A person can experience a maximum of 9 positive
G’s before blacking out.
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Some say that a human can experience up to 100 G’s for a
fraction of a second (the body compresses a little so it
doesn’t feel an instantaneous acceleration).
Roller coasters typically peak at 5 G’s to ensure safety and
comfort.
At >20 G’s there is a potential for death due to internal
injuries (organs are moving around).
A person can experience a maximum of 2-3
negative G’s before blood vessels in the eyes
rupture. Beyond 2-3 results in brain hemorrhage,
and other internal disturbances (marked discomfort).